In this paper, we aim to construct double Wronskian solutions for the FL equation (REF ).
This will allow us to have more freedom to understand possible distributions of eigenvalues,
present different kinds of solutions
(e.g. solitons, breathers and multiple pole solutions), explore their interactions
(cf.[1]}, [2]}),
and as a result, give a full profile of the FL equations
from bilinear approach and double Wronskian forms.
We will start from the pKN\((-1)\)  system ().
First we will bilinearize () and prove their double Wronskian solutions.
Note that the double Wronskians that we employ in the paper have different structures from those of the
AKNS hierarchy (cf.[3]}), the KN equation (cf.[4]})
and the Chen-Lee-Liu equation (cf.[5]}).
After that we will impose reductions on the double Wronskians using the technique
developed in [3]}.
This allows us to have solutions not only for the FL equation (REF )
but also for its nonlocal partner
\(u_{xt}+u-2i\delta uu(-x,-t)u_{x}=0,~~ \delta =\pm 1,\) 
